By a space we will always mean an object in this category, unless we
explicitly state that the space is “non-generalized”. It was observed by
Lawvere [1]} that it can alternatively be defined as the
category of categories and functors enriched over the posetal category
\(=[0,\infty ]\)  (with an arrow \(x\rightarrow y\)  whenever \(x\geqslant y\) ) equipped with the
monoidal structure induced by addition.
Our main interest in considering this category instead of the full subcategory
of (usual) metric spaces is double. Firstly, removing conditions (3) and (4)
makes the category much more well-behaved (the category of generalized metric
spaces has small colimits, which is not the case for metric spaces). Of course,
we generally work with separated (\(T_0\) ) spaces
and we will have to show that colimits we obtain from such spaces are still
separated. Secondly, removing condition (5) allows us to encode a notion of
direction of the space: intuitively, a point \(x\)  is really “before” a
point \(y\)  whenever the distance \(d(x,y)\)  is finite and \(d(y,x)=\infty \) .
